Adaptive weighting for flexible estimation in nonparametric regression models.

We propose a new method to find spatially adaptive smoothing splines. This new method breaks down the interval [0, 1] into p disjoint sub-intervals. Then we define p functional components in [0, 1], which have two importantfeatures. First, the purpose of each of these p components is to estimate the true function locally, i.e., in only one of the sub-intervals. Second, even though all components are defined on the entire domain, i.e. [0, 1], a component has curvature only inone of the aforementioned intervals. The p local estimates are then added togetherto produce a function estimate over the entire [0, 1] interval. In the proposed method, the additional flexibilitythat comes from finding these p local functional estimates does not come at any additional computational cost. In spite of having p components there is no need to specify (e.g., choose via cross validation) p smoothing parameters. Theory fromCOmponent Selection and Shrinkage Operator (COSSO), reduces the problem of specifying these p smoothing parameters to specifying only one smoothing parameterwithout a loss in flexibility. In fact, empirical studies indicate superior performance of COSSO in the additive model framework over that for the traditional additivemodel.